Zariski topology over multiplication Krasner hypermodules
UDC 512.5 In this paper, we introduce the notion of multiplication Krasner hypermodules over commutative hyperrings and topologize the collection of all multiplication Кrasner hypermodules. In addition, we investigate some properties of this topological space.
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| author | Kulak , Ö. Türkmen , B. N. Kulak , Ö. Türkmen , B. N. |
| author_facet | Kulak , Ö. Türkmen , B. N. Kulak , Ö. Türkmen , B. N. |
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| description | UDC 512.5
In this paper, we introduce the notion of multiplication Krasner hypermodules over commutative hyperrings and topologize the collection of all multiplication Кrasner hypermodules. In addition, we investigate some properties of this topological space. |
| doi_str_mv | 10.37863/umzh.v74i4.6626 |
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DOI: 10.37863/umzh.v74i4.6626
UDC 512.5
Ö. Kulak, B. N. Türkmen (Amasya Univ., Turkey)
ZARISKI TOPOLOGY OVER MULTIPLICATION KRASNER HYPERMODULES
ТОПОЛОГIЯ ЗАРИСЬКОГО НАД МУЛЬТИПЛIКАТИВНИМИ
ГIПЕРМОДУЛЯМИ КРАСНЕРА
In this paper, we introduce the notion of multiplication Krasner hypermodules over commutative hyperrings and topologize
the collection of all multiplication Кrasner hypermodules. In addition, we investigate some properties of this topological
space.
Мета цiєї роботи — визначення поняття мультиплiкативних гипермодулiв Краснера над комутативними гiперкiльця-
ми й топологiзацiя колекцiї всiх мультиплiкативних гипермодулiв Краснера, а також вивчення деяких властивостей
цього топологiчного простору.
1. Introduction. Notions of hypergroups, hypermodules and hyperrings have many important
roles in hyperstructures. Some authors have gotten many conclusions about these theories (see
[2, 7, 8, 10, 13]). It may also be eligible for reference [20] certain information about the theory of
rings and modules.
We recall some definitions and propositions from above references which we need to develop our
paper.
In this paper, we use \circ : M \times M \rightarrow P \ast (M) instead of \cdot : M \times M \rightarrow M, where M is a non-
empty set and P \ast (M) the set of all non-empty subsets of M. The map \circ is called a hyperoperation
on M. Thus, we use X \circ Y =
\bigcup
x\in X , y\in Y x \circ y, m \circ X = \{ m\} \circ X and X \circ m = X \circ \{ m\} for
all m \in M and X,Y \in P \ast (M) . The hyperstructure (M, o) is called a semihypergroup if, for all
x, y, z of M, we have (x \circ y) \circ z = x \circ (y \circ z) . A semihypergroup (M, \circ ) is called a hypergroup if,
for all m \in M, m \circ M = M \circ m = M [6].
A non-empty subset N of a hypergroup (M, \circ ) is called subhypergroup if, for all n \in N, we
have n \circ N = N \circ n = N. A hypergroup (M, \circ ) is called commutative if x \circ y = y \circ x for all
x, y \in M. A commutative hypergroup (M, \circ ) is said to be canonical, if there exists a unique 0 \in M
such that for all m \in M, m \circ 0 = \{ m\} , for all m \in M, there exists a unique m - 1 \in M such that
0 \in m \circ m - 1; if x \in y \circ z, then y \in xoz - 1 and z \in y - 1 \circ x for all x, y, z \in M [6].
The triple (R,\uplus , \circ ) is called a hyperring, if (R,\uplus ) is a semihypergroup, (R, o) is a semihyper-
group and \circ is a distributive over \uplus [8]. A Krasner hyperring is an algebraic structure (R,\uplus , \circ )
which satisfies the following axioms:
(1) (R,\uplus ) is a canonical hypergroup;
(2) (R, \circ ) is a semigroup having zero as a bilaterally absorbing element, i.e., x \circ 0 = 0 \circ x = 0;
(3) the multiplication is distributive with respect to the hyperoperation \uplus .
A Krasner hyperring (R,+, .) is called commutative (with unit element) if (R, .) is a commu-
tative semigroup (with unit element) [8]. A Krasner hyperring R is called a Krasner hyperfield, if
(R \setminus \{ 0\} , .) is a group [8]. Let A and B be non-empty subsets of a hyperring R. The sum A + B
is defined by A + B = \{ x | x \in a + b for some a \in A, b \in B \} . The product AB is defined
c\bigcirc Ö. KULAK, B. N. TÜRKMEN, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4 525
526 Ö. KULAK, B. N. TÜRKMEN
by AB =
\Bigl\{
x | x \in
\sum n
i=1
aibi, ai \in A, bi \in B, n \in \BbbZ +
\Bigr\}
. If A and B are hyperideals of R, then
A + B and AB are also hyperideals of R. Let (R,+, .) be a ring and I be a subset of R. G is
called a multiplicative subgroup of R if and only if (I, .) is a group. Moreover, if I is such that
R = RI and rG = Gr for every r \in R, then I is called a normal subgroup of R. As a normal
subgroup I of R induces an equivalence relation P in R and a partition of R in equivalence classes
which inherits a hyperring structure from R. Hyperrings obtained via this construction are called
quotient hyperrings and are denoted by R/I [8]. A non-empty subset I of a hyperring R is called
a hyperideal if (I,\uplus ) is a subhypergroup of (R,\uplus ) and r \circ a \cup a \circ r \subseteq I for all a \in I, r \in R [8].
A non-empty subset I of a Krasner hyperring (R,\uplus , \circ ) is called a left hyperideal of R if (I,\uplus ) is a
canonical subhypergroup of (R,\uplus ) and for all a \in I and r \in R, r \circ a \in I [19]. The hyperring R is
said to be commutative, if R is commutative with respect to hyperoperation “\cdot ”. A hyperring R has
identity element 1, if, for all r \in R, it is satisfied that r \in 1 \cdot r.
Let (R,\uplus , \circ ) be a hyperring and (M,+) be a hypergroup. If there exists an external hyper-
operation \cdot : R \times M \rightarrow P \ast (M) such that, for all a, b \in M and r, s \in R, we have r \cdot (a+ b) =
= (r \cdot a)+ (r \cdot b) and (r \circ s) \cdot a = r \cdot (s \cdot a) , then (M,+, \cdot ) is called a left hypermodule over R [4].
Similarly, a right hypermodule over R can be defined. M is called a hypermodule over R, if it is both
left and right hypermodule over R. If (M,+) is a canonical hypergroup and (R,\uplus , \circ ) is a Krasner
hyperring, then M is said to be a canonical R-hypermodule. In addition, M is called Krasner
R-hypermodule, if it is a canonical R-hypermodule, where “\cdot ” is an external operation, that is \cdot :
R\times M \rightarrow M by (r,m) \rightarrow r \cdot m \in M, and m \cdot 0 = 0. A non-empty subset N of an R-hypermodule
M is called a subhypermodule, if N is a hypermodule over R. A hypermodule M is called unitary if
1\cdot m = m for all m \in M. In this work, all R-hypermodules are left unitary Krasner R-hypermodules
unless otherwise stated.
Throughout this work, we admit that every hypermodule M is a Krasner R-hypermodule thereby
\{ 0\} is a subhypermodule of M. We denote a subhypermodule N of M by N \leq M. It can be seen
that for a Krasner R-hypermodule M and N \leq M, we can construct the quotient Krasner R-
hypermodule M/N, endowed with (a+N) \oplus (b+N) = \{ c+N : c \in a+ b\} and r \odot (a+N) =
= r \cdot a +N for all a +N, b +N \in M
N
and r \in R. A hypermodule M is said to be generated by
X such that M = \cap Y for all subhypermodules X \subset Y \subset M. If X is finite set, then a hypermodule
M is called a finitely generated hypermodule.
Let M1 and M2 be two R-hypermodules and f : M1 \rightarrow M2 be a function. Then f is called a
homomorphism if f (a+ b) \subseteq f (a) + f (b) and f (ra) = r \cdot f (a) for all a, b \in M1 and r \in R.
Also, f is called a strong homomorphism if f (a+ b) = f (a) + f (b) and f (ra) = r \cdot f (a) for
all a, b \in M1 and r \in R. In this case, we have f (0M1) = 0M2 . If a strong homomorphism f is
one-to-one and surjective function, it is called a strong isomorphism [11].
A subhypermodule N of M is called a maximal subhypermodule if there is no proper subhyper-
module of M contains N [1].
The Zariski topology was introduced primarily by Oskar Zariski. The Zariski topology is very
different from the usual Euclidean topology on \BbbR n or \BbbC n, which is defined by open sets with a union
or finite intersection of basic open sets which are balls. In the usual Euclidean topology, the ball with
radius \varepsilon > 0 centered at x \in \BbbA n is
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
ZARISKI TOPOLOGY OVER MULTIPLICATION KRASNER HYPERMODULES 527
B(x, r) =
\Biggl\{
y \in \BbbA n |
n\sum
i=1
| yi - xi| < \varepsilon
\Biggr\}
,
where \BbbA n is an affine space. Let \BbbF be an algebraically closed field and \BbbF [X] = \BbbF [X1, . . . , Xn] the
polynomial ring in n variables over \BbbF . A subset
Z = \{ y \in \BbbA n | f\alpha \in \BbbF [X] \forall \alpha \in A \} \subset \BbbA n
is called an algebraic set, if it is the set of common zeros of some collection of polynomials. It
also is written by Z = Z(\{ f\alpha \} \alpha \in A). For a single nonconstant f \in \BbbF [X], the set Z(f) is called a
hypersurface in \BbbA n. In Zariski topology, open sets are obtained by the complements of hypersurfaces
that are said principal open sets:
D(f) = \BbbA n - Z(f) = \{ y \in \BbbA n | f(y) \not = 0 \} .
In other words, a topology is defined on An and it is called a Zariski topology. If Z is any algebraic
set, then the Zariski topology on Z is the topology induced on it from \BbbA n. Closed (open) sets in \BbbA n
are intersections of \BbbA n with closed (open) sets in \BbbA n [18]. The Zariski topology is not Hausdorff; the
Zariski closed sets in \BbbA 1 are the empty set, finite collections of points in affine line and \BbbA 1 itself. If
\BbbF is infinite, the separation property of Hausdorff spaces fails when we meet any two nonempty open
sets. As it turns out, thanks to this topology, tools from topology are used to study algebraic varieties
which are the central objects of study in algebraic geometry. When there is an algebraic variety
on complex numbers, the Zariski topology is coarser than the usual topology, since every algebraic
set is closed for the usual topology [12]. In the literature, the Zariski topology was generalized for
structures such as the spectrum of a ring.
In [16] a topology on X = \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (M) is called the Zariski topology, in which closed sets are
varieties V (N) = \{ P \in X : P \supseteq N\} for all submodules N of M. We transfer this topology to hy-
permodules structures, we construct Zariski topology on X = \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (M) , where M is a hypermodule
and it consists of closed sets are varieties V (N) = \{ P \in X : P \supseteq N\} for all subhypermodules N of
M. In other words, we obtain that the Zariski topology is a specific construction equipping \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (M)
with a topology. In [9, 15 – 17] the definition of Zariski topology on prime spectrum of a module M
is different.
In this paper, we define the concept of multiplication hypermodules over a commutative hyperring
with identity and we topologize the spectrum of multiplication hypermodules and investigate the
properties of the induced topology. In addition we generalize the known results of Zariski topology
on \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (R) to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (M) . We prove that \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (M) is a T0 space and it is compact if and only if
M is finitely generated. In particular, we study cooperation between \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (M) and R/\mathrm{A}\mathrm{n}\mathrm{n}(N) and
obtain some important results.
2. Multiplication hypermodules. In this section we obtain respectable properties of multiplica-
tion hypermodules. These results will be used frequently in the next section.
Definition 2.1. (i) We call a unitary hypermodule M a multiplication R-hypermodule over
a commutative with identity hyperring R, if for every subhypermodule N of M, there exists a
hyperideal I of R such that N = I \cdot M.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
528 Ö. KULAK, B. N. TÜRKMEN
(ii) We call a proper subhypermodule N of an R-hypermodule M a prime hypermodule if
r \cdot a \in N for all r \in R and a \in M then either r \cdot M \subseteq N or a \in N. We will show that if N
is a prime subhypermodule of M then P = \mathrm{A}\mathrm{n}\mathrm{n} (M \setminus N) is necessarily a prime hyperideal of R.
We concentrate upon the notion of prime subhypermodules and a new kind of Zariski topology on
Spec(M) , prime spectrum of a hypermodule M, containing the set of all prime subhypermodules
of M.
(iii) Let M be an R-hypermodule and N be a subhypermodule of M such that N = I \cdot M for
some hyperideal I of R. Then, we call I a presentation hyperideal of N, or for short a presentation
of N. Here, it is not always true that a hypermodule N has presentation. So it is easily proven
that every subhypermodule of M has a presentation hyperideal if and only if M is a multiplication
hypermodule.
Now let us continue the section by classifying the properties of multiplication hypermodule.
Proposition 2.1. Let M be a non-zero multiplication R-hypermodule. Then every proper sub-
hypermodule of M is contained in a maximal subhypermodule of M.
Proof. Let N be any proper subhypermodule of M. By the hypothesis, there exists an ideal I
of R such that N = I \cdot M. We define a set \Psi of ideals in which all elements contain I. By Zorn’s
lemma, there exists a maximal element P of \Psi . So, N = I \cdot M \subseteq P \cdot M. It is shown easily that
P \cdot M is a maximal subhypermodule M.
Proposition 2.2. Let M be a non-zero multiplication R-hypermodule. Then N is a maximal
subhypermodule of M if and only if there exists a maximal hyperideal P of R such that N =
= P \cdot M \not = M.
Proof. (\Rightarrow ) Clear by Proposition 2.1.
(\Leftarrow ) Let U be a subhypermodule of M contains N . Since M is multiplication R-hypermodule,
there exists an ideal I of R such that U = I \cdot M. Since N \subseteq U, P \cdot M \subseteq I \cdot M and using maximality,
we have P = I. Hence, N is a maximal subhypermodule of M.
Theorem 2.1. The following statements are equivalent for a proper subhypermodule N of a
multiplication R-hypermodule M :
(i) N is a prime subhypermodule of M ;
(ii) \mathrm{A}\mathrm{n}\mathrm{n} (M \setminus N) is a prime hyperideal of R;
(iii) N = P \cdot M for some prime hyperideal P of R with \mathrm{A}\mathrm{n}\mathrm{n} (M) \subseteq P.
Proof. (i) \Rightarrow (ii) Let N be a prime subhypermodule of M. It is easily shown that \mathrm{A}\mathrm{n}\mathrm{n} (M \setminus N) =
= \{ a \in R : a \cdot x = 0 for all x \in M \setminus N\} is a hyperideal of R. Since N is a prime subhypermodule
of M, either a \cdot M \subseteq N or x \in N for all a \in R and x \in N. So, a \cdot x = 0 for all x \in M \setminus N,
a \in \mathrm{A}\mathrm{n}\mathrm{n} (M \setminus N) . Hence, \mathrm{A}\mathrm{n}\mathrm{n} (M \setminus N) is a prime hyperideal of R.
(ii) \Rightarrow (iiii) Let \mathrm{A}\mathrm{n}\mathrm{n} (M \setminus N) be a prime hyperideal of R. Suppose that N = P \cdot M for
a hyperideal P of R. So a \cdot M \subseteq N for all a \in P. Consider the hyperideal \mathrm{A}\mathrm{n}\mathrm{n} (M) =
= \{ a \in R : a \cdot M = \{ 0\} \} of R. By the definition, \mathrm{A}\mathrm{n}\mathrm{n} (M) \subseteq P and P = \mathrm{A}\mathrm{n}\mathrm{n} (M \setminus N) . There-
fore, P is a prime hyperideal of R.
(iii) \Rightarrow (i) Let N = P \cdot M for some prime hyperideal P of R with \mathrm{A}\mathrm{n}\mathrm{n} (M) \subseteq P. Let a \cdot x \in N
for all a \in R and x \in M. Since N = P \cdot M, then a \in M, a \cdot M \subseteq P \cdot M = N. Therefore, N is a
prime subhypermodule of M.
Proposition 2.3. Let N be a proper subhypermodule of a multiplication R-hypermodule M.
Then N is a prime subhypermodule of M if and only if AB \subseteq N implies A \subseteq N or B \subseteq N, where
A and B are subhypermodules of M.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
ZARISKI TOPOLOGY OVER MULTIPLICATION KRASNER HYPERMODULES 529
Proof. (\Rightarrow ) Let N be a prime subhypermodule of M and A \cdot B \subseteq N, but A \nsubseteq N and B \nsubseteq N
for some subhypermodules A and B of M. Suppose that I and J are presentations of A and B,
respectively. Then we have A \cdot B = I \cdot (J \cdot M) \subseteq N. Hence, there exist elements s \cdot x \in A \setminus N and
s\prime \cdot x\prime \in A \setminus N for some s \in I and s\prime \in J . Note that s \cdot (s\prime \cdot x\prime ) \in N. Hence, s \cdot M \subseteq N, that is,
s \cdot x \in N, which is a contradition.
(\Leftarrow ) Let r \cdot x \in N for some r \in R and x \in M \setminus N, but r \cdot M \nsubseteq N. Then r,m /\in H for some
m \in M. Let I and J be presentation hyperideals of rx and m, respectively. Then
R \cdot (r \cdot x) (R \cdot m) = (R \cdot x) (R \cdot r \cdot m) = (I \cdot m) (J \cdot m) = I \cdot (J \cdot M) = (I \circ J) \cdot M \subseteq N .
By the hypothesis, we have R \cdot x \subseteq N or R \cdot r \cdot m \subseteq N. It follows that x \in P or r \cdot x \in N, which
is a contradiction. Hence, N is a prime subhypermodule of M.
Corollary 2.1. Let N be a proper subhypermodule of M. Then N is a prime subhypermodule
of M if and only if whenever x, x
\prime \in M, xx\prime \subseteq N implies x \in N or x\prime \in N.
Proof. (\Rightarrow ) If N is a prime subhypermodule of M, it follows from xx\prime \subseteq N that x \in N or
x\prime \in N for every x, x\prime \in M .
(\Leftarrow ) Suppose that xx\prime \subseteq N implies x \in N or x\prime \in N, where x, x\prime \in M, and AB \subseteq N for
subhypermodules A and B of M, but A \nsubseteq N and B \nsubseteq N. Thus there exist elements a \in A \setminus N
and b \in B \setminus N. Then ab \in (R \cdot a) (R \cdot b) \subseteq AB \subseteq N and, hence, a \in A or b \in B, which is a
contradiction. Therefore, N is a prime subhypermodule of M.
3. Prime spectrum. In this section, we present the relationship of Zariski topology and multi-
plicative hypermodules. In order for authors to better understand this concept, we need the following
definition.
Definition 3.1. We denote by \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) the collection of all prime subhypermodules of M. It is
not always true that \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) is not be empty for any hypermodule M. We call such a hypermodule
primeless. It is clear that zero hypermodule is primeless. Now, we give a non-trival example.
Example 3.1. (i) Consider the Krasner hyperring \BbbZ and a multiplicatively closed subset S of \BbbZ
such that 0 /\in S. The equivalence relation \sim is defined on the set \BbbZ \times S as follows: (a, s) \sim (b, t)
if and only if there exists u \in S such that uta = usb. The equivalence class of (a, s) is denoted
by a/s and S - 1\BbbZ be the quotient set. By using [8] (Example 3.1.3 (10)), we see that S - 1\BbbZ is a
hyperring taking the hyperoperation of addition and multiplication. By [5] (Proposition 2.11), S - 1P
is a prime subhypermodule of S - 1\BbbZ where P = \langle p\rangle , p is a prime integer. So, \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (S - 1\BbbZ ) is
non-trivial. Therefore the hypermodule S - 1\BbbZ is primeless.
(ii) Consider the Prüfer group M = \BbbZ (p\infty ) for a prime integer p. Here the hypermodule M is
\BbbZ -hypermodule. Since
\biggl\langle
1
pn
+ \BbbZ
\biggr\rangle
is a prime subhypermodule of M for every integer n, \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M)
is non-trival.
(iii) Let R = K[\{ Xn\} \infty n=1] be a hyperring where K is a hyperfield and I = (X2
i , Xi - XiXj | j >
> i \geq 1), R = R/I and = (\{ Xn\} \infty n=1)/I. Then it can be seen in a similarly way from [3] (Example
2.5) that M is a multiplication R-hypermodule.
We denote by V \ast (N) the set of all prime subhypermodule of M containing N. Here V \ast (N) is
just the empty set and V \ast (\{ 0M\} ) is \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
530 Ö. KULAK, B. N. TÜRKMEN
Let I be any index set and Ni be any subhypermodules of M. Then note that
\bigcap
i\in I
V \ast (Ni) =
= V \ast
\Bigl( \sum
i\in I
Ni
\Bigr)
. Thus, if \xi \ast (M) denotes the collection of all subsets V \ast (N) of \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) ,
then \xi \ast (M) contains the empty set and \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) and is closed under arbitrary intersection. If
also \xi \ast (M) is closed under finite union, i.e., for any subhypermodules H and N of M such
that V \ast (H) \cup V \ast (N) = V \ast (J) , in this case \xi \ast (M) satisfies the axioms of closed subsets of a
topological space, which is Zariski topology.
Now we define a hypermodule with Zariski topology is called a top\ast hypermodule in the following
theorem and investigate the interesting properties of this topology. Firstly, we need the following
definition.
Definition 3.2. (i) We call the subhypermodule N of M a semiprime if N is as an intersection
of prime subhypermodules of M, and the subhypermodule L of M an extraordinary if whenever
U and V are semiprime subhypermodules of M with U \cap V \subseteq L, then U \subseteq L or V \subseteq L, and
the element m of a R-hypermodule M a unit provided that m is not contained in any maximal
subhypermodule of M.
(ii) We call the set r(N) = \{ x \in M | \exists n \in \BbbZ + : xn \in N \} radical hypermodule of N such that
r(N) is a subhypermodule of hypermodule M.
Theorem 3.1. Let M be an R-hypermodule. Then the following statements hold:
(i) M is a top\ast hypermodule;
(ii) every prime subhypermodule of M is extraordinary;
(iii) V \ast (N) \cup V \ast (L) = V \ast (N \cap L) for some semiprime subhypermodules N, and L of M.
Proof. Theorem can be proved similarly way in [17] (Lemma 2.1).
Proposition 3.1. Let M be a multiplication R-hypermodule. Then m \in M is unit if and only if
\langle m\rangle = M.
Proof. (\Leftarrow ) Is clear.
(\Rightarrow ) Suppose that an element m of M is unit. Then m is not contain in any maximal subhyper-
module of M. So, every proper subhypermodule of M is contained in a maximal subhypermodule,
a contradiction. Thus, we must have \langle m\rangle = M, by Proposition 2.1.
The following theorem shows that the Zariski topology is the topological space whose fundamen-
tal set is \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) .
Theorem 3.2. Let M be a multiplication R-hypermodule. Then the following statements hold
for a subset X of M, for subhypermodules N, L of M and, for every i \in I, a subhypermodule Ni
of M :
(i) V \ast (X) = V \ast (\langle X\rangle ) ;
(ii) V \ast (N) \cup V \ast (L) = V \ast (NL) = V \ast (N \cap L) ;
(iii)
\bigcap
i\in I
V \ast (Ni) = V \ast
\Bigl( \sum
i\in I
Ni
\Bigr)
;
(iv) V \ast (N) = V \ast (r (N)) ;
(v) if V \ast (N) \subseteq V \ast (L) , then L \subseteq r (N) ;
(vi) V \ast (N) = V \ast (L) if and only if r (N) = r (L) ;
(vii) V \ast (N) =
\bigcup
P\in V \ast R(Ann(M/N))
\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast P (M) , where
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
ZARISKI TOPOLOGY OVER MULTIPLICATION KRASNER HYPERMODULES 531
\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast P (M) = \{ P \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) : \mathrm{A}\mathrm{n}\mathrm{n} (M/P ) = p\} .
Proof. (i) Since X \subseteq \langle X\rangle , we have V \ast (X) = V \ast (\langle X\rangle ) . The converse follows that \langle X\rangle is a
smallest hypermodule contains X of M.
(ii) Since NL \subseteq N and NL \subseteq L, then we have V \ast (N) \cup V \ast (L) \subseteq V \ast (NL) . Conversely,
suppose that P \in V \ast (NL) . It follows from NL \subseteq P that N \subseteq P or L \subseteq P by Proposition 2.3.
Then P \in V \ast (N)\cup V \ast (L) . Thus, V \ast (NL) \subseteq V \ast (N)\cup V \ast (L) . So, V \ast (N)\cup V \ast (L) = V \ast (NL) .
The second equality immediately follows from Theorem 3.1 (iii).
(iii) It is clear that
\bigcap
i\in I
V \ast (Ni) \supseteq V \ast
\Bigl( \sum
i\in I
Ni
\Bigr)
. For the converse inclusion, suppose that
P \in V \ast
\Bigl( \sum
i\in I
Ni
\Bigr)
. Then
\sum
i\in I
Ni \subseteq P. So, we have Ni \subseteq P for every i \in I. Therefore,
P \in V \ast (Ni) for every i \in I. Conversely, V \ast
\Bigl( \sum
i\in I
Ni
\Bigr)
\subseteq
\bigcap
i\in I
V \ast (Ni) .
(iv) It is clear that H \subseteq r (H) . Then we have V \ast (r (N)) \subseteq V \ast (N). The converse inclusion is
obvious.
(v) It is clear.
(vi) It follows from (iii).
(vii) Suppose that P \in V \ast (N) . Since N \subseteq P, \mathrm{A}\mathrm{n}\mathrm{n}(M/N) \subseteq \mathrm{A}\mathrm{n}\mathrm{n}(M/P ) = P. It follows
from Propositions 2.1 and 2.2 that P \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast P (M) for P \in V \ast R (\mathrm{A}\mathrm{n}\mathrm{n} (M/N)). Conversely,
suppose that P \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast P (M) for P \in V \ast R (\mathrm{A}\mathrm{n}\mathrm{n} (M/N)) . Then we have \mathrm{A}\mathrm{n}\mathrm{n} (M/N) \subseteq P and
P = \mathrm{A}\mathrm{n}\mathrm{n} (M/N) for some P \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) . Since N = \mathrm{A}\mathrm{n}\mathrm{n} (M/N)M \subseteq \mathrm{A}\mathrm{n}\mathrm{n} (M/P ) = P, we
get P \in V \ast (N) .
Corollary 3.1. Every multiplication hypermodule is extraordinary.
For each subset S of M, by D(N) or X\ast
N we mean X - V \ast (N) . In particular, if S = \{ a\} ,
we denote D\ast (a) by D\ast
a or X\ast
a . Here, the sets X\ast
a are open, and they are called basic open sets and
X = \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) .
Theorem 3.3. Let M be a multiplication R-hypermodule. Then the following statements are
hold:
(i) X\ast
a \cap X\ast
b = X\ast
ab;
(ii) D\ast (I \cdot M) = D\ast (J \cdot M) = D\ast (I \cdot (J \cdot M)) for every hyperideal I and J of R;
(iii) X\ast
a = \varnothing \leftrightarrow a is nilpotent;
(iv) X\ast
a = X \leftrightarrow a is unit in M.
Proof. (i) It follows from Theorem 3.2 (ii).
(ii) By Proposition 2.3.
(iii) It is obtained using Corollary 2.1.
(iv) Clear by Propositions 2.1, 2.2 and 3.1.
In the following proposition, we have obtained a basis in the Zariski topology to built on multi-
plicative hypermodules.
Proposition 3.2. Let M be a multiplication R-hypermodule. Then the sets \{ X\ast
a | a \in M \}
form a basis for the Zariski topology on \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) .
Proof. Suppose that A is an open set in X = \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\ast (M) . Then there exists a subhyper-
module N of M such that A = X \setminus V \ast (N) . Let \{ ai : i \in I\} be a generator set of N, that
is, N = \langle \{ ai : i \in I\} \rangle . It follows from Theorem 3.2 (iii) that V \ast (N) = V \ast (\langle \{ ai : i \in I\} \rangle ) =
= V \ast
\Bigl( \sum
i\in I
R \cdot ai
\Bigr)
=
\bigcap
i\in I
V \ast (ai) . Thus, A = X \setminus V \ast (N) = X \setminus
\bigcap
i\in I
V \ast (ai) =
\bigcup
i\in I
X (ai) .
Consequently, \{ X\ast
a : a \in M\} is a basis for the Zariski topology on X.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
532 Ö. KULAK, B. N. TÜRKMEN
Compactness is a very important property of a topological space. For this reason, we will present
following theorem and proposition that give compactness properties.
Theorem 3.4. Let M be a multiplication R-hypermodule. Then every basic open set of X is
compact.
Proof. It is enough to show that every cover of basic open sets has a finite subcover. Suppose
that D\ast (a) \subseteq
\bigcup
i\in I
D\ast (ai) , and let N be the subhypermodule of M generated by ai’s. Note
that V \ast (N) =
\bigcap
i\in I
V \ast (ai) \subseteq D\ast (a) . By Theorem 3.2 (iv) and Theorem 3.1 (iii), V \ast (r (N)) \subseteq
\subseteq V \ast (r (\langle a\rangle )) , and so r (\langle a\rangle ) \subseteq r (N) . In addition, N =
\surd
B \cdot M, where B = \mathrm{A}\mathrm{n}\mathrm{n} (M/N) . Then
a =
\sum
\lambda \in \Lambda
s\lambda \cdot a\lambda , s\lambda \in
\surd
B , for a finite subset \Lambda of I. For s\lambda \in
\surd
B, we have s\ell \lambda \lambda \in B for some
positive integer \ell i. Let \ell =
\sum
\lambda \in \Lambda
\ell \lambda , then s\ell \lambda \in B for every \lambda \in \Lambda , that is, R \cdot a\lambda = I\lambda \cdot M. It
follows from a =
\sum
\lambda \in \Lambda
s\lambda \cdot a\lambda such that a \in
\sum
\lambda \in \Lambda
s\lambda \cdot I\lambda \cdot M =
\Bigl( \sum
\lambda \in \Lambda
(s\lambda \cdot I\lambda )
\Bigr)
\cdot M. Therefore,
a\ell \subseteq
\Bigl( \sum
\lambda \in \Lambda
s\lambda \cdot I\lambda
\Bigr) \ell
\cdot M \subseteq B \cdot M. Since
\bigcap
i\in I
V \ast (ai) = V \ast (N) \subseteq V \ast \bigl( a\ell \bigr) = V \ast (a) , we have
X\ast
a \subseteq
\bigcup
i\in I
X\ast (fi) .
Corollary 3.2. Let M be a multiplication R-hypermodule. An open set X is compact if and only
if it is finite union of basic open sets.
Proof. It follows from Proposition 3.2 and Theorem 3.4.
Proposition 3.3. Let M be a multiplication R-hypermodule. Then M is finitely generated if
and only if X is compact.
Proof. (\Rightarrow ) Since M is finitely generated, then M = \langle a1, a2, . . . , an\rangle . Then we have V \ast (\langle a1,
a2, . . . , an\rangle ) = \varnothing . Hence, D\ast (a1, a2, . . . , an) = X. Since
n\bigcup
i=1
X\ast
ai = X, X is compact.
(\Leftarrow ) Since X is compact, then X =
n\bigcup
i=1
X\ast
ai by Corollary 3.2. Then we have V \ast (\langle a1, a2, . . .
. . . , an\rangle ) =
n\bigcap
i=1
V \ast (ai) = \varnothing . It follows from Proposition 2.1 that M = \langle a1, a2, . . . , an\rangle . So, M
is finitely generated.
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Received 14.03.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
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| id | umjimathkievua-article-6626 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:29:26Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/60/708f9b3e8ba53e6f7fa5bf6d3755d160.pdf |
| spelling | umjimathkievua-article-66262022-07-06T16:22:31Z Zariski topology over multiplication Krasner hypermodules Zariski topology over multiplication Krasner hypermodules Kulak , Ö. Türkmen , B. N. Kulak , Ö. Türkmen , B. N. мультиплiкативні гипермодулi Краснера Multiplication Krasner Hypermodules, Prime Subhypermodules, Spectrum of Krasner Hypermodules, Zariski Topology. UDC 512.5 In this paper, we introduce the notion of multiplication Krasner hypermodules over commutative hyperrings and topologize the collection of all multiplication Кrasner hypermodules. In addition, we investigate some properties of this topological space. УДК 512.5 Мета цiєї роботи — визначення поняття мультиплiкативних гипермодулiв Краснера над комутативними гiперкiльцями й топологiзацiя колекцiї всiх мультиплiкативних гипермодулiв Краснера, а також вивчення деяких властивостей цього топологiчного простору. Institute of Mathematics, NAS of Ukraine 2022-05-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6626 10.37863/umzh.v74i4.6626 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 4 (2022); 525 - 533 Український математичний журнал; Том 74 № 4 (2022); 525 - 533 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6626/9221 Copyright (c) 2022 Burcu NISANCI TURKMEN |
| spellingShingle | Kulak , Ö. Türkmen , B. N. Kulak , Ö. Türkmen , B. N. Zariski topology over multiplication Krasner hypermodules |
| title | Zariski topology over multiplication Krasner hypermodules |
| title_alt | Zariski topology over multiplication Krasner hypermodules |
| title_full | Zariski topology over multiplication Krasner hypermodules |
| title_fullStr | Zariski topology over multiplication Krasner hypermodules |
| title_full_unstemmed | Zariski topology over multiplication Krasner hypermodules |
| title_short | Zariski topology over multiplication Krasner hypermodules |
| title_sort | zariski topology over multiplication krasner hypermodules |
| topic_facet | мультиплiкативні гипермодулi Краснера Multiplication Krasner Hypermodules Prime Subhypermodules Spectrum of Krasner Hypermodules Zariski Topology. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6626 |
| work_keys_str_mv | AT kulako zariskitopologyovermultiplicationkrasnerhypermodules AT turkmenbn zariskitopologyovermultiplicationkrasnerhypermodules AT kulako zariskitopologyovermultiplicationkrasnerhypermodules AT turkmenbn zariskitopologyovermultiplicationkrasnerhypermodules |